The human rights bill, which has bipartisan support in both houses of Congress, is named after Sergei Magnitsky, a lawyer who died in prison in 2009 after being prosecuted on charges that his supporters argue were manufactured to cover up official corruption.

The divisions over Russia as an issue in the American campaign came starkly into focus as Mrs. Clinton wrapped up a 10-day, six-nation trip to Asia that overlapped with the national party conventions at home.

Mr. Putin and other Russian officials have vehemently opposed the Magnitsky bill, warning of so-far-unspecified reciprocal measures if it is enacted into law.

Although Mrs. Clinton did not mention the Magnitsky legislation in public on Saturday, a senior State Department official said she raised the issue of human rights and recent moves by Russia to restrict nongovernmental organizations during a private breakfast of crepes and red caviar with Foreign Minister Sergey V. Lavrov.

The Jackson-Vanik trade barriers were imposed to punish the Soviet Union for its restrictions on the emigration of Jews. Although the Jackson-Vanik provisions have been waived since the Soviet Union fell apart, their continued existence in American law would allow Russia to maintain higher duties on American products of its own choosing.

The administration has had tensions with Russia this year over Mr. Putin’s re-election, the protests that followed and the government’s intensifying crackdown on opponents, including three members of the punk band Pussy Riot, who were sentenced to two years in prison for a political protest in a cathedral. The United States has also been upset by Russia’s refusal to support efforts intended to force President Bashar al-Assad from power in Syria to try to stop the war there.

The senior State Department official said that Mrs. Clinton and Mr. Putin also discussed Syria and efforts to halt Iran’s nuclear program, as well as plans by the United States for a missile defense system in Europe that Russia opposes. The two then sat next to each other at dinner and chatted throughout the 90-minute meal about wildlife conservation; the Russian Far East; the Winter Olympics to be held in Sochi, Russia, in 2014; and other topics, the official said.

Mrs. Clinton and Mr. Lavrov announced modest agreements on noncontentious issues and sought to portray them as examples of what Mr. Lavrov called “constructive cooperation.”

“Is it possible to find a solution to the problem, if current President Obama is re-elected for a second term? Theoretically, yes,” Mr. Putin said, according to the official transcript posted on the Kremlin’s Web site. “But this isn’t just about President Obama.

“For all I know, his desire to work out a solution is quite sincere,” Mr. Putin continued. “I met him recently on the sidelines of the G-20 summit in Los Cabos, Mexico, where we had a chance to talk. And though we talked mostly about Syria, I could still take stock of my counterpart. My feeling is that he is a very honest man, and that he sincerely wants to make many good changes. But can he do it? Will they let him do it?”

He added a sharp rebuke, accusing Mr. Romney of using inflamed language for political gain.

That criticism might seem curious coming from Mr. Putin. His own campaign for office made use of some pointed anti-American talk, including criticism of the missile plan, claims that Secretary of State Hillary Rodham Clinton had sent “a signal” to prompt antigovernment demonstrations after a disputed parliamentary election, and accusations that the American ambassador, Michael A. McFaul, was meddling in Russian domestic affairs.